A logical propositional portion common to a prior art and the present invention and necessary to understand both of them will be described. Generally known mathematical basic knowledge will be described first.
A cosine wave P·cos(ω·t) and a sine wave Q·sin(ω·t) which have the same frequency but different amplitudes are combined into the following cosine wave. Let P and Q be amplitudes, and ω be an angular frequency.P·cos(ω·t)+Q·sin(ω·t)=(P2+Q2)1/2·cos(ω·t−ε) for ε=tan−1(Q/P)  (1)
In order to analyze the combining operation in equation (1), it is convenient to perform mapping on a complex coordinate plane so as to plot an amplitude P of cosine wave P·cos(ω·t) along a real axis and an amplitude Q of the sine wave Q·sin(ω·t) along an imaginary axis. That is, on the complex coordinate plane, a distance (P2+Q2)1/2 from the origin gives the amplitude of the combined wave, and an angle e=tan−1(Q/P) gives the phase difference between the combined wave and ω·t.
In addition, on the complex coordinate plane, the following relational expression holds.L·exp(j·ε)=L·cos(ε)+j·L·sin(ε)  (2)
Equation (2) is an expression associated with a complex vector, in which j is an imaginary unit, L gives the length of the complex vector, and e gives the direction of the complex vector. In order to analyze the geometrical relationship on the complex coordinate plane, it is convenient to use conversion to a complex vector.
The following description uses mapping onto a complex coordinate plane like that described above and geometrical analysis using complex vectors to show how an inter-electrode electromotive force behaves and explain how the prior art uses this behavior.
A complex vector arrangement with one coil set and an electrode pair in the electromagnetic flowmeter proposed by the present inventor (see patent reference WO 03/027614) will be described next.
FIG. 25 is a block diagram for explaining the principle of the electromagnetic flowmeter in the above patent reference. This electromagnetic flowmeter includes a measuring tube 1 through which a fluid to be measured flows, a pair of electrodes 2a and 2b which are placed to face each other in the measuring tube 1 so as to be perpendicular to both a magnetic field to be applied to the fluid and an axis PAX of the measuring tube 1 and come into contact with the fluid, and detect the electromotive force generated by the magnetic flow and the flow of the fluid, and an exciting coil 3 which applies, to the fluid, a time-changing magnetic field asymmetric on the front and rear sides of the measuring tube 1 which are bordered on a plane PLN which includes the electrodes 2a and 2b, with the plane PLN serving as a boundary of the measuring tube 1.
Of a magnetic field Ba generated by the exciting coil 3, a magnetic field component (magnetic flux density) B1 orthogonal to both an electrode axis EAX connecting the electrodes 2a and 2b and the measuring tube axis PAX on the electrode axis EAX is given byB1=b1·cos(ω0·t−θ1)  (3)
In equation (3), b1 is the amplitude of the magnetic flux density B1, ω0 is an angular frequency, and θ1 is a phase difference (phase lag) from ω0·t. The magnetic flux density B1 will be referred to as the magnetic field B1 hereinafter.
An inter-electrode electromotive force which originates from a change in magnetic field and is irrelevant to the flow velocity of a fluid to be measured will be described first. Since the electromotive force originating from the change in magnetic field depends on a time derivative dB/dt of the magnetic field, and hence the magnetic field B1 generated by the exciting coil 3 is differentiated according todB1/dt=−ω0·b1·sin(ω0·t−θ1)  (4)
If the flow velocity of the fluid to be measured is 0, a generated eddy current is only a component originating from a change in magnetic field. An eddy current I due to a change in the magnetic field Ba is directed as shown in FIG. 26. Therefore, an inter-electrode electromotive force E which is generated by a change in the magnetic field Ba and is irrelevant to the flow velocity is directed as shown in FIG. 26 within a plane including the electrode axis EAX and the measuring tube axis PAX. This direction is defined as the negative direction.
At this time, the inter-electrode electromotive force E is the value obtained by multiplying a time derivative −dB1/dt of a magnetic field whose direction is taken into consideration by a coefficient k (a complex number associated with the conductivity and permittivity of the fluidity to be measured and the structure of the measuring tube 1 including the layout of the electrodes 2a and 2b), as indicated by the following equation:E=k·ω0·b1·sin(ω0·t−θ1)  (5)
Equation (5) is rewritten into the following equation:
                                                                           E                =                                ⁢                                                                            k                      ·                      ω                                        ⁢                                                                                  ⁢                                          0                      ·                                              b                        ⁢                        1                                            ·                                              {                                                  sin                          ⁡                                                      (                                                                                          -                                θ                                                            ⁢                                                                                                                          ⁢                              1                                                        )                                                                          }                                            ·                      cos                                        ⁢                                          (                                              ω                        ⁢                                                                                                  ⁢                                                  0                          ·                          t                                                                    )                                                        +                                                                                                                        ⁢                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                              cos                        ⁡                                                  (                                                                                    -                              θ                                                        ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                        ·                    sin                                    ⁢                                      (                                          ω                      ⁢                                                                                          ⁢                                              0                        ·                        t                                                              )                                                                                                                          =                                ⁢                                                                            k                      ·                      ω                                        ⁢                                                                                  ⁢                                          0                      ·                                              b                        ⁢                        1                                            ·                                              {                                                  -                                                      sin                            ⁡                                                          (                                                              θ                                ⁢                                                                                                                                  ⁢                                1                                                            )                                                                                                      }                                            ·                      cos                                        ⁢                                          (                                              ω                        ⁢                                                                                                  ⁢                                                  0                          ·                          t                                                                    )                                                        +                                                                                                                        ⁢                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                              cos                        ⁡                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                        ·                    sin                                    ⁢                                      (                                          ω                      ⁢                                                                                          ⁢                                              0                        ·                        t                                                              )                                                  ⁢                                                                                                                            (          6          )                    
In this case, if equation (6) is mapped on the complex coordinate plane with reference to ω0·t, a real axis component Ex and an imaginary axis component Ey are given by
                                                                           Ex                =                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                                                        -                          sin                                                ⁢                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                                                                                                                              =                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                              cos                        ⁡                                                  (                                                                                    π                              2                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                                                )                                                                    }                                                                                                                                (          7          )                                                                                        Ey                =                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                              cos                        ⁡                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                                                                                                                              =                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                                          {                                              sin                        ⁡                                                  (                                                                                    π                              2                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                                                )                                                                    }                                                                                                                                (          8          )                    
In addition, Ex and Ey represented by equations (7) and (8) are transformed into a complex vector Ec represented by
                                                                           Ec                =                                ⁢                                                      Ex                    +                                          j                      ·                      Ey                                                        =                                                                                    k                        ·                        ω                                            ⁢                                                                                          ⁢                                              0                        ·                                                  b                          ⁢                          1                                                ·                        ·                                                  {                                                      cos                            ⁡                                                          (                                                                                                π                                  2                                                                +                                                                  θ                                  ⁢                                                                                                                                          ⁢                                  1                                                                                            )                                                                                }                                                                                      +                                                                                                                                          ⁢                                                      j                    ·                    k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ⁢                                                                                  ·                                          {                                              sin                        ⁡                                                  (                                                                                    π                              2                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                                                )                                                                    }                                                                                                                                                                =                                    ⁢                                                                                    k                        ·                        ω                                            ⁢                                                                                          ⁢                                              0                        ·                                                  b                          ⁢                          1                                                ·                                                  {                                                      cos                            ⁡                                                          (                                                                                                π                                  2                                                                +                                                                  θ                                  ⁢                                                                                                                                          ⁢                                  1                                                                                            )                                                                                }                                                                                      +                                          j                      ·                                              sin                        ⁡                                                  (                                                                                    π                              2                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                                                )                                                                                                                    }                                                                                        =                                ⁢                                                      k                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              π                            2                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                                          )                                                              }                                                                                                            (          9          )                    
In addition, the coefficient k described above is transformed into a complex vector to obtain the following equation:
                                                                           k                =                                                      rk                    ·                                          cos                      ⁡                                              (                                                  θ                          ⁢                                                                                                          ⁢                          00                                                )                                                                              +                                      j                    ·                    rk                    ·                                          sin                      ⁡                                              (                                                  θ                          ⁢                                                                                                          ⁢                          00                                                )                                                                                                                                                                    =                                  rk                  ·                                      exp                    ⁡                                          (                                                                        j                          ·                          θ                                                ⁢                                                                                                  ⁢                        00                                            )                                                                                                                                (          10          )                    
In equation (10), rk is a proportional coefficient, and θ00 is the angle of the vector k with respect to the real axis.
Substituting equation (10) into equation (9) yields an inter-electrode electromotive force Ec (an inter-electrode electromotive force which originates from only a temporal change in magnetic field and is irrelevant to the flow velocity) transformed into complex coordinates as follows:
                                                                           Ec                =                                                      rk                    ·                                          exp                      ⁡                                              (                                                                              j                            ·                            θ                                                    ⁢                                                                                                          ⁢                          00                                                )                                                              ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              π                            2                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                                          )                                                              }                                                                                                                          =                                                      rk                    ·                    ω                                    ⁢                                                                          ⁢                                      0                    ·                                          b                      ⁢                      1                                        ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              π                            2                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            00                                                                          )                                                              }                                                                                                            (          11          )                    
In equation (11), rk·ω0·b1·exp{j·(p/2+θ1+θ00)} is a complex vector having a length rk·ω0·b1 and an angle p/2+θ1+θ00 with respect to the real axis.
An inter-electrode electromotive force originating from the flow velocity of a fluid to be measured will be described next. Letting V (V≠0) be the magnitude of the flow velocity of the fluid, since a component v×Ba originating from a flow velocity vector v of the fluid is generated in a generated eddy current in addition to the eddy current I when the flow velocity is 0, an eddy current Iv generated by the flow velocity vector v and the magnetic field Ba is directed as shown in FIG. 27. Therefore, the direction of an inter-electrode electromotive force Ev generated by the flow velocity vector v and the magnetic field Ba becomes opposite to the direction of the inter-electrode electromotive force E generated by the temporal change, and the direction of Ev is defined as the positive direction.
In this case, as indicated by the following equation, the inter-electrode electromotive force Ev originating from the flow velocity is the value obtained by multiplying the magnetic field B1 as indicated by the following equation by a coefficient kv (a complex number associated with a magnitude V of the flow velocity, the conductivity and permittivity of the fluidity to be measured, and the structure of the measuring tube 1 including the arrangement of the electrodes 2a and 2b):Ev=kv·{b1·cos(ω0·tθ1)}  (12)
Equation (12) is rewritten into
                                                        Ev              =                            ⁢                                                                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          cos                      ⁡                                              (                                                  ω                          ⁢                                                                                                          ⁢                                                      0                            ·                            t                                                                          )                                                              ·                                          cos                      ⁡                                              (                                                  -                          θ1                                                )                                                                                            -                                                      kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          sin                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                              ·                                                                                                                                        ⁢                              sin                ⁡                                  (                                      -                    θ1                                    )                                                                                                        =                            ⁢                                                                    kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              cos                        ⁡                                                  (                          θ1                          )                                                                    }                                        ·                                          cos                      ⁡                                              (                                                  ω0                          ·                          t                                                )                                                                                            +                                                      kv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                                          {                                              sin                        ⁢                                                  (                          θ1                          )                                                                    }                                        ·                                                                                                                                        ⁢                              sin                ⁡                                  (                                      ω0                    ·                    t                                    )                                                                                        (        13        )            
In this case, when mapping equation (13) on the complex coordinate plane with reference to ω0·t, a real axis component Evx and an imaginary axis component Evy are given by
                                 Evx          =                      kv            ·                          b              ⁢              1                        ·                          {                              cos                ⁡                                  (                                      θ                    ⁢                                                                                  ⁢                    1                                    )                                            }                                                            (          14          )                                              Evy          =                      kv            ·                          b              ⁢              1                        ·                          {                              sin                ⁡                                  (                                      θ                    ⁢                                                                                  ⁢                    1                                    )                                            }                                                            (          15          )                    
In addition, Evx and Evy represented by equations (14) and (15) are transformed into a complex vector Evc represented by
                                                                           Evc                =                                  Evx                  +                                      j                    ·                    Evy                                                                                                                          =                                                      kv                    ·                                          b                      ⁢                      1                                        ·                                          {                                              cos                        ⁡                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                                        +                                      j                    ·                    kv                    ·                                          b                      ⁢                      1                                        ⁢                                                                                  ·                                          {                                              sin                        ⁡                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    }                                                                                                                                              =                                  k                  ·                                      b                    ⁢                    1                                    ·                                      {                                                                  cos                        ⁡                                                  (                                                      θ                            ⁢                                                                                                                  ⁢                            1                                                    )                                                                    +                                              j                        ·                                                  sin                          ⁡                                                      (                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                        )                                                                                                                }                                                                                                                          =                                                      k                    ·                                          b                      ⁢                      1                                        ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                  θ                          ⁢                                                                                                          ⁢                          1                                                )                                                              }                                                                                                            (          16          )                    
In addition, the coefficient kv described above is transformed into a complex vector to obtain the following equation:
                                                                           kv                =                                                      rkv                    ·                                          cos                      ⁡                                              (                                                  θ                          ⁢                                                                                                          ⁢                          01                                                )                                                                              +                                      j                    ·                    rkv                    ·                                          sin                      ⁡                                              (                                                  θ                          ⁢                                                                                                          ⁢                          01                                                )                                                                                                                                                                    =                                  rkv                  ·                                      exp                    ⁡                                          (                                                                        j                          ·                          θ                                                ⁢                                                                                                  ⁢                        01                                            )                                                                                                                                (          17          )                    
In equation (17), rkv is a proportional coefficient, and θ01 is the angle of the vector kv with respect to the real axis. In this case, rkv is equivalent to the value obtained by multiplying the proportional coefficient rk (see equation (10)) described above by the magnitude V of the flow velocity and a proportion coefficient γ. That is, the following equation holds:rkv=γ·rk·V  (18)
Substituting equation (17) into equation (16) yields an inter-electrode electromotive force Evc transformed into complex coordinates as follows:
                                                                           Evc                =                                  kv                  ·                                      b                    ⁢                    1                                    ·                                      exp                    ⁡                                          (                                                                        j                          ·                          θ                                                ⁢                                                                                                  ⁢                        1                                            )                                                                                                                                              =                                                      rkv                    ·                                          b                      ⁢                      1                                        ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              θ                            ⁢                                                                                                                  ⁢                            1                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            01                                                                          )                                                              }                                                                                                            (          19          )                    
In equation (19), rkv·b1·exp{j·(θ1+θ01)} is a complex vector having a length rkv·b1 and an angle θ1+θ01 with respect to the real axis.
An inter-electrode electromotive force Eac as a combination of inter-electrode electromotive force Ec originating from a temporal change in magnetic field and an inter-electrode electromotive force Evc originating from the flow velocity of the fluid is expressed by the following equation according to equations (11) and (19).
                                                                           Eac                =                                ⁢                                  Ec                  +                  Evc                                                                                                        =                                ⁢                                                                            rk                      ·                      ω                                        ⁢                                                                                  ⁢                                          0                      ·                      b                                        ⁢                                                                                  ⁢                                          1                      ·                      exp                                        ⁢                                          {                                              j                        ·                                                  (                                                                                    π                              /                              2                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              1                                                        +                                                          θ                              ⁢                                                                                                                          ⁢                              00                                                                                )                                                                    }                                                        +                                                                                                                        ⁢                                                      rkv                    ·                    b                                    ⁢                                                                          ⁢                                      1                    ·                    exp                                    ⁢                                      {                                          j                      ·                                              (                                                                              θ                            ⁢                                                                                                                  ⁢                            1                                                    +                                                      θ                            ⁢                                                                                                                  ⁢                            01                                                                          )                                                              }                                                                                                            (          20          )                    
As is obvious from equation (20), an inter-electrode electromotive force Eac is written by two complex vectors rk·ω0·b1·exp{j·(p/2+θ1+θ00)} and rkv·b1·exp{j·(θ1+θ01)}. The length of the resultant vector obtained by combining the two complex vectors represents the amplitude of the output (the inter-electrode electromotive force Eac), and an angle φ of the resultant vector represents the phase difference (phase delay) of the inter-electrode electromotive force Eac with respect to the phase ω0·t of the input (exciting current). Note that a flow rate is obtained by multiplying a flow velocity by the cross-sectional area of the measuring tube. In general, therefore, a flow velocity and a flow rate have a one-to-one relationship in calibration in an initial state, and obtaining a flow velocity is equivalent to obtaining a flow rate. For this reason, the following description will exemplify the scheme of obtaining a flow velocity (for obtaining a flow rate).
Under the above principle, the electromagnetic flowmeter in the above patent reference extracts a parameter (asymmetric excitation parameter) free from the influence of a span shift, and outputs a flow rate on the basis of the extracted parameter, thereby solving the problem of the span shift.
A span shift will be described with reference to FIG. 28. Assume that the magnitude V of the flow velocity measured by the electromagnetic flowmeter has changed in spite of the fact that the flow velocity of a fluid to be measured has not changed. In such a case, a span shift can be thought as a cause of this output variation.
Assume that calibration is performed such that when the flow velocity of a fluid to be measured is 0 in an initial state, the output from the electromagnetic flowmeter becomes 0 (v), and when the flow velocity is 1 (m/sec), the output becomes 1 (v). In this case, an output from the electromagnetic flowmeter is a voltage representing the magnitude V of a flow velocity. According to this calibration, if the flow velocity of a fluid to be measured is 1 (m/sec), the output from the electromagnetic flowmeter should be 1 (v). When a given time t1 has elapsed, however, the output from the electromagnetic flowmeter may become 1.2 (v) in spite of the fact that the flow velocity of the fluid to be measured remains 1 (m/sec). A span shift can be thought as a cause of this output variation. A phenomenon called a span shift occurs when, for example, the value of an exciting current flowing in the exciting coil cannot be maintained constant.